Introduction to post-quantum cryptography

Introduction to post-quantum cryptography Daniel J. Bernstein Department of Computer Science, University of Illinois at Chicago. 1 Is cryptography dead? Imagine that it s fifteen years from now and someone announces the successful construction of a large quantum computer. The New York Times runs a frontpage article reporting that all of the public-key algorithms used to protect the Internet have been broken. Users panic. What exactly will happen to cryptography? Perhaps, after seeing quantum computers destroy RSA and DSA and ECDSA, Internet users will leap to the conclusion that cryptography is dead; that there is no hope of scrambling information to make it incomprehensible to, and unforgeable by, attackers; that securely storing and communicating information means using expensive physical shields to prevent attackers from seeing the information for example, hiding USB sticks inside a locked briefcase chained to a trusted courier s wrist. A closer look reveals, however, that there is no justification for the leap from quantum computers destroy RSA and DSA and ECDSA to quantum computers destroy cryptography. There are many important classes of cryptographic systems beyond RSA and DSA and ECDSA: Hash-based cryptography. The classic example is Merkle s hash-tree public-key signature system (1979), building upon a one-message-signature idea of Lamport and Diffie. Code-based cryptography. The classic example is McEliece s hidden- Goppa-code public-key encryption system (1978). Lattice-based cryptography. The example that has perhaps attracted the most interest, not the first example historically, is the Hoffstein Pipher Silverman NTRU public-key-encryption system (1998). Multivariate-quadratic-equations cryptography. One of many interesting examples is Patarin s HFE v public-key-signature system (1996), generalizing a proposal by Matsumoto and Imai.

2 Daniel J. Bernstein Secret-key cryptography. The leading example is the Daemen Rijmen Rijndael cipher (1998), subsequently renamed AES, the Advanced Encryption Standard. All of these systems are believed to resist classical computers and quantum computers. Nobody has figured out a way to apply Shor s algorithm the quantum-computer discrete-logarithm algorithm that breaks RSA and DSA and ECDSA to any of these systems. Another quantum algorithm, Grover s algorithm, does have some applications to these systems; but Grover s algorithm is not as shockingly fast as Shor s algorithm, and cryptographers can easily compensate for it by choosing somewhat larger key sizes. Is there a better attack on these systems? Perhaps. This is a familiar risk in cryptography. This is why the community invests huge amounts of time and energy in cryptanalysis. Sometimes cryptanalysts find a devastating attack, demonstrating that a system is useless for cryptography; for example, every usable choice of parameters for the Merkle Hellman knapsack public-key encryption system is easily breakable. Sometimes cryptanalysts find attacks that are not so devastating but that force larger key sizes. Sometimes cryptanalysts study systems for years without finding any improved attacks, and the cryptographic community begins to build confidence that the best possible attack has been found or at least that real-world attackers will not be able to come up with anything better. Consider, for example, the following three factorization attacks against RSA: 1978: The original paper by Rivest, Shamir, and Adleman mentioned a new algorithm, Schroeppel s linear sieve, that factors any RSA modulus n and thus breaks RSA using 2 (1+o(1))(lg n)1/2 (lg lg n) 1/2 simple operations. Here lg = log 2. Forcing the linear sieve to use at least 2 b operations means choosing n to have at least (0.5+o(1))b 2 /lg b bits. Warning: 0.5+o(1) means something that converges to 0.5 as b.it does not say anything about, e.g., b = 128. Figuring out the proper size of n for b = 128 requires looking more closely at the speed of the linear sieve. 1988: Pollard introduced a new factorization algorithm, the number-field sieve. This algorithm, as subsequently generalized by Buhler, Lenstra, and Pomerance, factors any RSA modulus n using 2 (1.9...+o(1))(lg n)1/3 (lg lg n) 2/3 simple operations. Forcing the number-field sieve to use at least 2 b operations means choosing n to have at least (0.016...+ o(1))b 3 /(lg b) 2 bits. Today, twenty years later, the fastest known factorization algorithms for classical computers still use 2 (constant+o(1))(lg n)1/3 (lg lg n) 2/3 operations. There have been some improvements in the constant and in the details of the o(1), but one might guess that 1/3 is optimal, and that choosing n to have roughly b 3 bits resists all possible attacks by classical computers.

Introduction to post-quantum cryptography 3 1994: Shor introduced an algorithm that factors any RSA modulus n using (lg n) 2+o(1) simple operations on a quantum computer of size (lg n) 1+o(1). Forcing this algorithm to use at least 2 b operations means choosing n to have at least 2 (0.5+o(1))b bits an intolerable cost for any interesting value of b. See the Quantum computing chapter of this book for much more information on quantum algorithms. Consider, for comparison, attacks on another thirty-year-old public-key cryptosystem, namely McEliece s hidden-goppa-code encryption system. The original McEliece paper presented an attack that breaks codes of length n and dimension n/2 using2 (0.5+o(1))n/lg n operations. Forcing this attack to use 2 b operations means choosing n at least (2+o(1))b lg b. Several subsequent papers have reduced the number of attack operations by an impressively large factor, roughly n lg n =2 (lg n)2, but (lg n) 2 is much smaller than 0.5n/lg n if n is large; the improved attacks still use 2 (0.5+o(1))n/lg n operations. One can reasonably guess that 2 (0.5+o(1))n/lg n is best possible. Quantum computers don t seem to make much difference, except for reducing the constant 0.5. If McEliece s cryptosystem is holding up so well against attacks, why are we not already using it instead of RSA? The answer, in a nutshell, is efficiency, specifically key size. McEliece s public key uses roughly n 2 /4 b 2 (lg b) 2 bits, whereas an RSA public key assuming the number-field sieve is optimal and ignoring the threat of quantum computers uses roughly (0.016...)b 3 /(lg b) 2 bits. If b were extremely large then the b 2+o(1) bits for McEliece would be smaller than the b 3+o(1) bits for RSA; but real-world security levels such as b = 128 allow RSA key sizes of a few thousand bits, while McEliece key sizes are closer to a million bits. Figure 1 summarizes the process of designing, analyzing, and optimizing cryptographic systems before the advent of quantum computers; Figure 2 summarizes the same process after the advent of quantum computers. Both pictures have the same structure: cryptographers design systems to scramble and unscramble data; cryptanalysts break some of those systems; algorithm designers and implementors find the fastest unbroken systems. Cryptanalysts in Figure 1 use the number-field sieve for factorization, the Lenstra Lenstra Lovasz algorithm for lattice-basis reduction, the Faugère algorithms for Gröbner-basis computation, and many other interesting attack algorithms. Cryptanalysts in Figure 2 have all of the same tools in their arsenal plus quantum algorithms, notably Shor s algorithm and Grover s algorithm. All of the most efficient unbroken public-key systems in Figure 1, perhaps not coincidentally, take advantage of group structures that can also be exploited by Shor s algorithm, so those systems disappear from Figure 2, and the users end up with different cryptographic systems.

4 Daniel J. Bernstein Cryptographers: How can we encrypt, decrypt, sign, verify, etc.? Cryptanalysts: What can an attacker do using < 2 b operations on a classical computer? Functioning cryptographic systems: DES, Triple DES, AES, RSA, McEliece encryption, Merkle hash-tree signatures, Merkle Hellman knapsack encryption, Buchmann Williams class-group encryption, ECDSA, HFE v, NTRU, etc. Unbroken cryptographic systems: Triple DES (for b 112), AES (for b 256), RSA with b 3+o(1) -bit modulus, McEliece with code length b 1+o(1), Merkle signatures with strong b 1+o(1) -bit hash, BW with strong b 2+o(1) -bit discriminant, ECDSA with strong b 1+o(1) -bit curve, HFE v with b 1+o(1) polynomials, NTRU with b 1+o(1) bits, etc. Algorithm designers and implementors: Exactly how small and fast are the unbroken cryptosystems? Users Most efficient unbroken cryptosystems: e.g., can verify signature in time b 2+o(1) using ECDSA with strong b 1+o(1) -bit curve Fig. 1. Pre-quantum cryptography. Warning: Sizes and times are simplified to b 1+o(1), b 2+o(1), etc. Optimization of any specific b requires a more detailed analysis; e.g., low-exponent RSA verification is faster than ECDSA verification for small b.

Introduction to post-quantum cryptography 5 Cryptographers: How can we encrypt, decrypt, sign, verify, etc.? Cryptanalysts: What can an attacker do using < 2 b operations on a quantum computer? Functioning cryptographic systems: DES, Triple DES, AES, RSA, McEliece encryption, Merkle hash-tree signatures, Merkle Hellman knapsack encryption, Buchmann Williams class-group encryption, ECDSA, HFE v, NTRU, etc. Unbroken cryptographic systems: AES (for b 128), McEliece with code length b 1+o(1), Merkle signatures with strong b 1+o(1) -bit hash, HFE v with b 1+o(1) polynomials, NTRU with b 1+o(1) bits, etc. Algorithm designers and implementors: Exactly how small and fast are the unbroken cryptosystems? Users Most efficient unbroken cryptosystems: e.g., can verify signature in time b 3+o(1) using HFE v with b 1+o(1) polynomials Fig. 2. Post-quantum cryptography. Warning: Sizes and times are simplified to b 1+o(1), b 2+o(1), etc. Optimization of any specific b requires a more detailed analysis.

6 Daniel J. Bernstein 2 A taste of post-quantum cryptography Here are three specific examples of cryptographic systems that appear to be extremely difficult to break even for a cryptanalyst armed with a large quantum computer. Two of the examples are public-key signature systems; one of the examples is a public-key encryption system. All three examples are parametrized by b, the user s desired security level. Many more parameters and variants appear later in this book, often allowing faster encryption, decryption, signing, and verification with smaller keys, smaller signatures, etc. I chose to focus on public-key examples a focus shared by most of this book because quantum computers seem to have very little effect on secretkey cryptography, hash functions, etc. Grover s algorithm forces somewhat larger key sizes for secret-key ciphers, but this effect is essentially uniform across ciphers; today s fastest pre-quantum 256-bit ciphers are also the fastest candidates for post-quantum ciphers at a reasonable security level. (There are a few specially structured secret-key ciphers that can be broken by Shor s algorithm, but those ciphers are certainly not today s fastest ciphers.) For an introduction to state-of-the-art secret-key ciphers I recommend the following book: Matthew Robshaw and Olivier Billet (editors), New stream cipher designs: the estream finalists, Lecture Notes in Computer Science 4986, Springer, 2008, ISBN 978 3 540 68350 6. 2.1 A hash-based public-key signature system This signature system requires a standard cryptographic hash function H that produces 2b bits of output. For b = 128 one could choose H as the SHA- 256 hash function. Over the last few years many concerns have been raised regarding the security of popular hash functions, and over the next few years NIST will run a competition for a SHA-256 replacement, but all known attacks against SHA-256 are extremely expensive. The signer s public key in this system has 8b 2 bits: e.g., 16 kilobytes for b = 128. The key consists of 4b strings y 1 [0],y 1 [1],y 2 [0],y 2 [1],...,y 2b [0],y 2b [1], each string having 2b bits. A signature of a message m has 2b(2b +1) bits: e.g., 8 kilobytes for b = 128. The signature consists of 2b-bit strings r, x 1,...,x 2b such that the bits (h 1,...,h 2b ) of H(r, m) satisfy y 1 [h 1 ]=H(x 1 ), y 2 [h 2 ]=H(x 2 ),andsoon through y 2b [h 2b ]=H(x 2b ). How does the signer find x with H(x) =y? Answer: The signer starts by generating a secret x and then computes y = H(x). Specifically, the signer s secret key has 8b 2 bits, namely 4b independent uniform random strings x 1 [0],x 1 [1],x 2 [0],x 2 [1],...,x 2b [0],x 2b [1], each string having 2b bits. The signer computes the public key y 1 [0],y 1 [1],y 2 [0],y 2 [1],...,y 2b [0],y 2b [1] as H(x 1 [0]),H(x 1 [1]),H(x 2 [0]),H(x 2 [1]),...,H(x 2b [0]),H(x 2b [1]).

Introduction to post-quantum cryptography 7 To sign a message m, the signer generates a uniform random string r, computes the bits (h 1,...,h 2b ) of H(r, m), andreveals(r, x 1 [h 1 ],...,x 2b [h 2b ]) as a signature of m. The signer then discards the remaining x values and refuses to sign any more messages. What I ve described so far is the Lamport Diffie one-time signature system. What do we do if the signer wants to sign more than one message? An easy answer is chaining. The signer includes, in the signed message, a newly generated public key that will be used to sign the next message. The verifier checks the first signed message, including the new public key, and can then check the signature of the next message; the signature of the nth message includes all n 1 previous signed messages. More advanced systems, such as Merkle s hash-tree signature system, scale logarithmically with the number of messages signed. To me hash-based cryptography is a convincing argument for the existence of secure post-quantum public-key signature systems. Grover s algorithm is the fastest quantum algorithm to invert generic functions, and is widely believed to be the fastest quantum algorithm to invert the vast majority of specific efficiently computable functions (although obviously there are also many exceptions, i.e., functions that are easier to invert). Hash-based cryptography can convert any hard-to-invert function into a secure public-key signature system. See the Hash-based digital signature schemes chapter of this book for a much more detailed discussion of hash-based cryptography. Note that most hash-based systems impose an extra requirement of collision resistance upon the hash function, allowing simpler signatures without randomization. 2.2 A code-based public-key encryption system Assume that b is a power of 2. Write n =4b lg b; d = lg n ; andt = 0.5n/d. For example, if b = 128, thenn = 3584; d =12;andt = 149. The receiver s public key in this system is a dt n matrix K with coefficients in F 2. Messages suitable for encryption are n-bit strings of weight t, i.e., n-bit strings having exactly t bits set to 1. To encrypt a message m, the sender simply multiplies K by m, producing a dt-bit ciphertext Km. The basic problem for the attacker is to syndrome-decode K, i.e., to undo the multiplication by K, knowing that the input had weight t. Itiseasy,by linear algebra, to work backwards from Km to some n-bit vector v such that Kv = Km; however, there are a huge number of choices for v, and finding a weight-t choice seems to be extremely difficult. The best known attacks on this problem take time exponential in b for most matrices K. How, then, can the receiver solve the same problem? The answer is that the receiver generates the public key K with a secret structure, specifically a hidden Goppa code structure, that allows the receiver to decode in a reasonable amount of time. It is conceivable that the attacker can detect the hidden Goppa code structure in the public key, but no such attack is known.

8 Daniel J. Bernstein Specifically, the receiver starts with distinct elements α 1,α 2,...,α n of the field F 2 d and a secret monic degree-t irreducible polynomial g F 2 d[x]. The main work for the receiver is to syndrome-decode the dt n matrix 1/g(α 1 ) 1/g(α n ) α 1 /g(α 1 ) α n /g(α n ) H =......., α1 t 1 /g(α 1 ) αn t 1 /g(α n ) where each element of F 2 d is viewed as a column of d elements of F 2 in a standard basis of F 2 d. This matrix H is a parity-check matrix for an irreducible binary Goppa code, and can be syndrome-decoded by Patterson s algorithm or by faster algorithms. The receiver s public key K is a scrambled version of H. Specifically,the receiver s secret key also includes an invertible dt dt matrix S and an n n permutation matrix P. The public key K is the product SHP. Given a ciphertext Km = SHPm, the receiver multiplies by S 1 to obtain HPm, decodes H to obtain Pm, and multiplies by P 1 to obtain m. What I ve described here is a variant, due to Niederreiter (1986), of McEliece s original code-based public-key encryption system. Both systems are extremely efficient at key generation, encryption, and decryption, but as I mentioned earlier have been held back by their long public keys. See the Code-based cryptography and Lattice-based cryptography chapters of this book for much more information about code-based cryptography and (similar but more complicated) lattice-based cryptography, including several systems that use shorter public keys. 2.3 A multivariate-quadratic public-key signature system The public key in this system is a sequence P 1,P 2,...,P 2b F 2 [w 1,...,w 4b ]: a sequence of 2b polynomials in the 4b variables w 1,...,w 4b, with coefficients in F 2 = {0, 1}. Each polynomial is required to have degree at most 2, withno squared terms, and is represented as a sequence of 1+4b +4b(4b 1)/2 bits, namely the coefficients of 1,w 1,...,w 4b,w 1 w 2,w 1 w 3,...,w 4b 1 w 4b. Overall the public key has 16b 3 +4b 2 +2b bits; e.g., 4 megabytes for b = 128. A signature of a message m has just 6b bits: namely, 4b values w 1,...,w 4b F 2 and a 2b-bit string r satisfying H(r, m) =(P 1 (w 1,...,w 4b ),...,P 2b (w 1,...,w 4b )). Here H is a standard hash function. Verifying a signature uses one evaluation of H and roughly b 3 bit operations to evaluate P 1,...,P 2b. The critical advantage of this signature system over hash-based signature systems is that each signature is short. Other multivariate-quadratic systems have even shorter signatures and, in many cases, much shorter public keys.

Introduction to post-quantum cryptography 9 The basic problem faced by an attacker is to find a sequence of 4b bits w 1,...,w 4b producing 2b specified output bits (P 1 (w 1,...,w 4b ),...,P 2b (w 1,...,w 4b )). Guessing a sequence of 4b bits is fast but has, on average, chance only 2 2b of success. More advanced equation-solving attacks, such as XL, can succeed in considerably fewer than 2 2b operations, but no known attacks have a reasonable chance of succeeding in 2 b operations for most quadratic polynomials P 1,...,P 2b in 4b variables. The difficulty of this problem is not surprising, given how general the problem is: every inversion problem can be rephrased as a problem of solving multivariate quadratic equations. How, then, can the signer solve the same problem? The answer, as in Section 2.2, is that the signer generates the public key P 1,...,P 2b with a secret structure, specifically an HFE v structure, that allows the signer to solve the equations in a reasonable amount of time. It is conceivable that the attacker can detect the HFE v structure in the public key, or in the public key together with a series of legitimate signatures; but no such attack is known. Fix a standard irreducible polynomial ϕ F 2 [t] of degree 3b. Define L as the field F 2 [t]/ϕ of size 2 3b. The critical step in signing is finding roots of a secret low-degree univariate polynomial over L: specifically, a polynomial in L[x] of degree at most 2b. There are several standard algorithms that do this in time b O(1). The secret polynomial is chosen to have all nonzero exponents of the form 2 i +2 j or 2 i.ifanelementx L is expressed in the form x 0 + x 1 t + + x 3b 1 t 3b 1,witheachx i F 2,thenx 2 = x 0 +x 1 t 2 + +x 3b 1 t 6b 2 and x 4 = x 0 +x 1 t 4 + +x 3b 1 t 12b 4 and so on, so x 2i +2 j is a quadratic polynomial in the variables x 0,...,x 3b 1. Some easy extra transformations hide the structure of this polynomial, producing the signer s public key. Specifically, the signer s secret key has three components: An invertible 4b 4b matrix S with coefficients in F 2. A polynomial Q L[x, v 1,v 2,...,v b ] where each term has one of the following six forms: lx 2i +2 j with l L, 2 i < 2 j, 2 i +2 j 2b; lx 2i v j with l L, 2 i 2b; lv i v j ; lx 2i ; lv j ; l. Ifb = 128 then there are 9446 possible terms, each having a 384-bit coefficient l, for a total of 443 kilobytes. A 2b 3b matrix T of rank 2b with coefficients in F 2. The signer computes the public key as follows. Compute a column vector (x 0,x 1,...,x 3b 1,v 1,v 2,...,v b ) as S times the column vector (w 1,...,w 4b ). Inside the quotient ring L[w 1,...,w 4b ]/(w 2 1 w 1,...,w 2 4b w 4b), compute x = x i t i and y = Q(x, v 1,v 2,...,v b ). Write y as y 0 + y 1 t + + y 3b 1 t 3b 1 with each y i in F 2 [w 1,...,w 4b ], and compute (P 1,P 2,...,P 2b ) as T times the column vector (y 0,y 1,...,y 3b 1 ). Signing works backwards through the same construction:

10 Daniel J. Bernstein Starting from the desired values of P 1,P 2,...,P 2b, solve the secret linear equations T (y 0,y 1,...,y 3b 1 )=(P 1,P 2,...,P 2b ) to obtain values of (y 0,y 1,...,y 3b 1 ).Thereare2 b possibilities for (y 0,y 1,...,y 3b 1 );choose one of those possibilities randomly. Choose values v 1,v 2,...,v b F 2 randomly, and substitute these values into the secret polynomial Q(x, v 1,v 2,...,v b ), obtaining a polynomial Q(x) L[x]. Compute y = y 0 +y 1 t+ +y 3b 1 t 3b 1 L, andsolveq(x) =y, obtaining x L. If there are several roots x of Q(x) = y, choose one of them randomly. If there are no roots, restart the signing process. Write x as x 0 + x 1 t + + x 3b 1 t 3b 1 with x 0,...,x 3b 1 F 2. Solve the secret linear equations S(w 1,...,w 4b )=(x 0,...,x 3b 1,v 1,...,v b ), obtaining a signature (w 1,...,w 4b ). This is an example of a class of HFE v constructions introduced by Patarin in 1996. HFE refers to the Hidden Field Equation Q(x) =y.the refers to the omission of some bits: Q(x) =y is equivalent to 3b equations on bits, but only 2b equations are published. The v refers to the vinegar variables v 1,v 2,...,v b. Pure HFE, with no omitted bits and no vinegar variables, is (lg b)2 breakable in time roughly 2 by Gröbner-basis attacks, but HFE v has solidly resisted attack for more than ten years. There are many other ways to build multivariate-quadratic public-key systems, and many interesting ideas for saving time and space, producing a huge number of candidates for post-quantum cryptography; see the Multivariate public key cryptography chapter of this book. It is hardly a surprise that some of the fastest candidates have been broken. A recent paper by Dubois, Fouque, Shamir, and Stern, after breaking an extremely simplified system with no vinegar variables and with only one nonzero term in Q, leaps to the conclusion that all multivariate-quadratic systems are dangerous: Multivariate cryptographic schemes are very efficient but have a lot of exploitable mathematical structure. Their security is not fully understood, and new attacks against them are found on a regular basis. It would thus be prudent not to use them in any security-critical applications. Presumably the same authors would recommend already avoiding 4096-bit RSA in a pre-quantum world since 512-bit RSA has been broken, would recommend avoiding all elliptic curves since a few special elliptic curves have been broken (clearly elliptic curves have a lot of exploitable mathematical structure ), and would recommend avoiding 256-bit AES since DES has been broken ( new attacks against ciphers are found on a regular basis ). My own recommendation is that the community continue to systematically study the security and efficiency of cryptographic systems, so that we can identify the highest-security systems that fit the speed and space requirements imposed by cryptographic users.

Introduction to post-quantum cryptography 11 3 Challenges in post-quantum cryptography Let me review the picture so far. Some cryptographic systems, such as RSA with a four-thousand-bit key, are believed to resist attacks by large classical computers but do not resist attacks by large quantum computers. Some alternatives, such as McEliece encryption with a four-million-bit key, are believed to resist attacks by large classical computers and attacks by large quantum computers. So why do we need to worry now about the threat of quantum computers? Why not continue to focus on RSA and ECDSA? If someone announces the successful construction of a large quantum computer fifteen years from now, why not simply switch to McEliece etc. fifteen years from now? This section gives three answers three important reasons that parts of the cryptographic community are already starting to focus attention on postquantum cryptography: We need time to improve the efficiency of post-quantum cryptography. We need time to build confidence in post-quantum cryptography. We need time to improve the usability of post-quantum cryptography. In short, we are not yet prepared for the world to switch to post-quantum cryptography. Maybe this preparation is unnecessary. Maybe we won t actually need post-quantum cryptography. Maybe nobody will ever announce the successful construction of a large quantum computer. However, if we don t do anything, and if it suddenly turns out years from now that users do need post-quantum cryptography, years of critical research time will have been lost. 3.1 Efficiency Elliptic-curve signature systems with O(b)-bit signatures and O(b)-bit keys appear to provide b bits of security against classical computers. State-of-theart signing algorithms and verification algorithms take time b 2+o(1). Can post-quantum public-key signature systems achieve similar levels of performance? My two examples of signature systems certainly don t qualify: one example has signatures of length b 2+o(1), and the other example has keys of length b 3+o(1). There are many other proposals for post-quantum signature systems, but I have never seen a proposal combining O(b)-bit signatures, O(b)- bit keys, polynomial-time signing, and polynomial-time verification. Inefficient cryptography is an option for some users but is not an option for a busy Internet server handling tens of thousands of clients each second. If you make a secure web connection today to https://www.google.com, Google redirects your browser to http://www.google.com, deliberately turning off cryptographic protection. Google does have some cryptographically protected web pages but apparently cannot afford to protect its most heavily used web pages. If Google already has trouble with the slowness of today s cryptographic

12 Daniel J. Bernstein software, surely it will not have less trouble with the slowness of post-quantum cryptographic software. Constraints on space and time have always posed critical research challenges to cryptographers and will continue to pose critical research challenges to post-quantum cryptographers. On the bright side, research in cryptography has produced many impressive speedups, and one can reasonably hope that increased research efforts in post-quantum cryptography will continue to produce impressive speedups. There has already been progress in several directions; for details, read the rest of this book! 3.2 Confidence Merkle s hash-tree public-key signature system and McEliece s hidden-goppacode public-key encryption system were both proposed thirty years ago and remain essentially unscathed despite extensive cryptanalytic efforts. Many other candidates for hash-based cryptography and code-based cryptography are much newer; multivariate-quadratic cryptography and latticebased cryptography provide an even wider variety of new candidates for postquantum cryptography. Some specific proposals have been broken. Perhaps a new system will be broken as soon as a cryptanalyst takes the time to look at the system. One could insist on using classic systems that have survived many years of review. But often the user cannot afford the classic systems and is forced to consider newer, smaller, faster systems that take advantage of more recent research into cryptographic efficiency. To build confidence in these systems the community needs to make sure that cryptanalysts have taken time to search for attacks on the systems. Those cryptanalysts, in turn, need to gain familiarity with post-quantum cryptography and experience with post-quantum cryptanalysis. 3.3 Usability The RSA public-key cryptosystem started as nothing more than a trapdoor one-way function, cube modulo n. (Tangential historical note: The original paper by Rivest, Shamir, and Adleman actually used large random exponents. Rabin pointed out that small exponents such as 3 are hundreds of times faster.) Unfortunately, one cannot simply use a trapdoor one-way function as if it were a secure encryption function. Modern RSA encryption does not simply cube a message modulo n; it has to first randomize and pad the message. Furthermore, to handle long messages, it encrypts a short random string instead of the message, and uses that random string as a key for a symmetric cipher to encrypt and authenticate the original message. This infrastructure around RSA took many years to develop, with many disasters along the way, such as the PKCS#1 v1.5 padding standard broken by Bleichenbacher in 1998.

Introduction to post-quantum cryptography 13 Furthermore, even if a secure encryption function has been defined and standardized, it needs software implementations and perhaps also hardware implementations suitable for integration into a wide variety of applications. Implementors need to be careful not only to achieve correctness and speed but also to avoid timing leaks and other side-channel leaks. A few years ago several implementations of RSA and AES were broken by cache-timing attacks; Intel has, as a partial solution, added AES instructions to its future CPUs. This book describes randomization and padding techniques for some postquantum systems, but much more work remains to be done. Post-quantum cryptography, like the rest of cryptography, needs complete hybrid systems and detailed standards and high-speed leak-resistant implementations. 4 Comparison to quantum cryptography Quantum cryptography, also called quantum key distribution, expands a short shared key into an effectively infinite shared stream. The prerequisite for quantum cryptography is that the users, say Alice and Bob, both know (e.g.) 256 unpredictable secret key bits. The result of quantum cryptography is that Alice and Bob both know a stream of (e.g.) 10 12 unpredictable secret bits that can be used to encrypt messages. The length of the output stream increases linearly with the amount of time that Alice and Bob spend on quantum cryptography. This description of quantum cryptography might make quantum cryptography sound like a synonym for stream cipher. The prerequisite for a stream cipher for example, counter-mode AES is that Alice and Bob both know (e.g.) 256 unpredictable secret key bits. The result of a stream cipher is that Alice and Bob both know a stream of (e.g.) 10 12 unpredictable secret bits that can be used to encrypt messages. The length of the output stream increases linearly with the amount of time that Alice and Bob spend on the stream cipher. However, the details of quantum cryptography are quite different from the details of a stream cipher: A stream cipher generates the output stream as a mathematical function of the input key. Quantum cryptography uses physical techniques for Alice to continuously generate random secret bits and to encode those bits for transmission to Bob. A stream cipher can be used to protect information sent through any number of untrusted hops on any existing network; eavesdropping fails because the encrypted information is incomprehensible. Quantum cryptography requires a direct fiber-optic connection between Alice s trusted quantumcryptography hardware and Bob s trusted quantum-cryptography hardware; eavesdropping fails because it interrupts the communication. Even if a stream cipher is implemented perfectly, its security is merely conjectural nobody has figured out an attack so we conjecture that no

14 Daniel J. Bernstein attack exists. If quantum cryptography is implemented perfectly then its security follows from generally accepted laws of quantum mechanics. A modern stream cipher can run on any commonly available CPU, and generates gigabytes of stream per second on a $200 CPU. Quantum cryptography generates kilobytes of stream per second on special hardware costing $50000. One can reasonably argue that quantum cryptography, locked-briefcase cryptography, meet-privately-in-a-sealed-vaultcryptography, and otherphysical shields for information are part of post-quantum cryptography: they will not be destroyed by quantum computers! But post-quantum cryptography is, in general, a quite different topic from quantum cryptography: Post-quantum cryptography, like the rest of cryptography, covers a wide range of secure-communication tasks, ranging from secret-key operations, public-key signatures, and public-key encryption to high-level operations such as secure electronic voting. Quantum cryptography handles only one task, namely expanding a short shared secret into a long shared secret. Post-quantum cryptography, like the rest of cryptography, includes some systems proven to be secure, but also includes many lower-cost systems that are conjectured to be secure. Quantum cryptography rejects conjectural systems begging the question of how Alice and Bob can securely share a secret in the first place. Post-quantum cryptography includes many systems that can be used for a noticeable fraction of today s Internet communication Alice and Bob need to perform some computation and send some data but do not need any new hardware. Quantum cryptography requires new network hardware that is, at least for the moment, impossibly expensive for the vast majority of Internet users. My own interests are in cryptographic techniques that can be widely deployed across the Internet; I see tremendous potential in post-quantum cryptography and very little hope for quantum cryptography. To be fair I should report the views of the proponents of quantum cryptography. Magiq, a company that sells quantum-cryptography hardware, has the following statement on its web site: Once the enormous energy boost that quantum computers are expected to provide hits the street, most encryption security standards and any other standard based on computational difficulty will fall, experts believe. Evidently these unnamed experts believe and Magiq would like you to believe that quantum computers will break AES, and dozens of other wellknown secret-key ciphers, and Merkle s hash-tree signature system, and McEliece s hidden-goppa-code encryption system, and Patarin s HFE v signature system, and NTRU, and all of the other cryptographic systems discussed in this book. Time will tell whether this belief was justified!

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